Step |
Hyp |
Ref |
Expression |
1 |
|
isinitoi.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
2 |
|
isinitoi.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
3 |
|
isinitoi.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
4 |
1 2 3
|
isinitoi |
⊢ ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) → ( 𝑂 ∈ 𝐵 ∧ ∀ 𝑜 ∈ 𝐵 ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑜 ) ) ) |
5 |
|
oveq2 |
⊢ ( 𝑜 = 𝑂 → ( 𝑂 𝐻 𝑜 ) = ( 𝑂 𝐻 𝑂 ) ) |
6 |
5
|
eleq2d |
⊢ ( 𝑜 = 𝑂 → ( ℎ ∈ ( 𝑂 𝐻 𝑜 ) ↔ ℎ ∈ ( 𝑂 𝐻 𝑂 ) ) ) |
7 |
6
|
eubidv |
⊢ ( 𝑜 = 𝑂 → ( ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑜 ) ↔ ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑂 ) ) ) |
8 |
7
|
rspcv |
⊢ ( 𝑂 ∈ 𝐵 → ( ∀ 𝑜 ∈ 𝐵 ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑜 ) → ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑂 ) ) ) |
9 |
8
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) ∧ 𝑂 ∈ 𝐵 ) → ( ∀ 𝑜 ∈ 𝐵 ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑜 ) → ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑂 ) ) ) |
10 |
|
eusn |
⊢ ( ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑂 ) ↔ ∃ ℎ ( 𝑂 𝐻 𝑂 ) = { ℎ } ) |
11 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
12 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) ∧ 𝑂 ∈ 𝐵 ) → 𝐶 ∈ Cat ) |
13 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) ∧ 𝑂 ∈ 𝐵 ) → 𝑂 ∈ 𝐵 ) |
14 |
1 2 11 12 13
|
catidcl |
⊢ ( ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) ∧ 𝑂 ∈ 𝐵 ) → ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ∈ ( 𝑂 𝐻 𝑂 ) ) |
15 |
|
fvex |
⊢ ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ∈ V |
16 |
15
|
elsn |
⊢ ( ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ∈ { ℎ } ↔ ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) = ℎ ) |
17 |
|
eqcom |
⊢ ( ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) = ℎ ↔ ℎ = ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ) |
18 |
|
sneqbg |
⊢ ( ℎ ∈ V → ( { ℎ } = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ↔ ℎ = ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ) ) |
19 |
18
|
bicomd |
⊢ ( ℎ ∈ V → ( ℎ = ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ↔ { ℎ } = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
20 |
19
|
elv |
⊢ ( ℎ = ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ↔ { ℎ } = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) |
21 |
16 17 20
|
3bitri |
⊢ ( ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ∈ { ℎ } ↔ { ℎ } = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) |
22 |
21
|
biimpi |
⊢ ( ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ∈ { ℎ } → { ℎ } = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) |
23 |
22
|
a1i |
⊢ ( ( 𝑂 𝐻 𝑂 ) = { ℎ } → ( ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ∈ { ℎ } → { ℎ } = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
24 |
|
eleq2 |
⊢ ( ( 𝑂 𝐻 𝑂 ) = { ℎ } → ( ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ∈ ( 𝑂 𝐻 𝑂 ) ↔ ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ∈ { ℎ } ) ) |
25 |
|
eqeq1 |
⊢ ( ( 𝑂 𝐻 𝑂 ) = { ℎ } → ( ( 𝑂 𝐻 𝑂 ) = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ↔ { ℎ } = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
26 |
23 24 25
|
3imtr4d |
⊢ ( ( 𝑂 𝐻 𝑂 ) = { ℎ } → ( ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) ∈ ( 𝑂 𝐻 𝑂 ) → ( 𝑂 𝐻 𝑂 ) = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
27 |
14 26
|
syl5 |
⊢ ( ( 𝑂 𝐻 𝑂 ) = { ℎ } → ( ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) ∧ 𝑂 ∈ 𝐵 ) → ( 𝑂 𝐻 𝑂 ) = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
28 |
27
|
exlimiv |
⊢ ( ∃ ℎ ( 𝑂 𝐻 𝑂 ) = { ℎ } → ( ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) ∧ 𝑂 ∈ 𝐵 ) → ( 𝑂 𝐻 𝑂 ) = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
29 |
28
|
com12 |
⊢ ( ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) ∧ 𝑂 ∈ 𝐵 ) → ( ∃ ℎ ( 𝑂 𝐻 𝑂 ) = { ℎ } → ( 𝑂 𝐻 𝑂 ) = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
30 |
10 29
|
syl5bi |
⊢ ( ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) ∧ 𝑂 ∈ 𝐵 ) → ( ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑂 ) → ( 𝑂 𝐻 𝑂 ) = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
31 |
9 30
|
syld |
⊢ ( ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) ∧ 𝑂 ∈ 𝐵 ) → ( ∀ 𝑜 ∈ 𝐵 ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑜 ) → ( 𝑂 𝐻 𝑂 ) = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
32 |
31
|
expimpd |
⊢ ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) → ( ( 𝑂 ∈ 𝐵 ∧ ∀ 𝑜 ∈ 𝐵 ∃! ℎ ℎ ∈ ( 𝑂 𝐻 𝑜 ) ) → ( 𝑂 𝐻 𝑂 ) = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) ) |
33 |
4 32
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑂 ∈ ( InitO ‘ 𝐶 ) ) → ( 𝑂 𝐻 𝑂 ) = { ( ( Id ‘ 𝐶 ) ‘ 𝑂 ) } ) |